Ring Homomorphism from $\mathbb{Z}$ to a Field $F$

Question

I have so far shown that $\phi: \mathbb{Z} \to F$ is a unique ring homomorphism if $\phi(n)=n$ and that $ker(\phi) = (n) = \{nm|m \in \mathbb{Z}\}$ with $n = 0$ or $n=p$ with $p$ a prime number. ($n$ is called the characteristic of $F$).

Now, I have to show the following two points.

1. If $F$ has characteristic $p \gt 0$, $\phi$ induces an injective homomorphism $\mathbb{F}_{p} \hookrightarrow F$. ($\mathbb{F}_{p}$ is a subfield of $F$).

2. Let $F$ be a finite field with $q$ elements. Show that $q$ is a positive power of the characteristic $p$ of $F$.

I just really have no clue how to approach these problems and would appreciate a lot if help is given. I am also not even sure what $\hookrightarrow$ means. Can anybody help? Thanks.

Answer 1

Hint: For 1, the first isomorphism theorem for rings describes how, in general, $\phi:R \to S$ induces an injective homomorphism from $R/\ker \phi$.

For 2, note that $\Bbb F_q$ is a vector space over its subfield $\Bbb F_p$.